A course I much enjoyed this semester was on the equations of mathematical physics (PHYS384 for those at SFU.) We learnt a whole bunch of methods to solve and analyze equations that fit into the Sturm-Liouville framework of linear partial differential equations. The natural next step is to extend this body of knowledge to non-linear equations of mathematical physics. That is exactly what I’m doing next term.
We have a whole bunch of methods to analyze and predict the performance of linear systems, but systems in nature aren’t always that idealistic. The classic example is that of a frictionless pendulum with the equation of motion:
g is the acceleration due to
l is the length of the pendulum. The sine function being
non-linear is often linearized to just for small
angles. This is the standard harmonic oscillator equation.
The real challenge is in analyzing the original non-linear equation. The difficulty in solving this equation is readily seen in the following Maxima session, where it isn’t smart enough to recognize the Jacobian elliptical integral.
The only course I’m taking next term is titled “Instabilities, Chaos and Turbulence in non-linear and complex systems” numbered PHYS484. We’ll explore a lot of these concepts. I’m really looking forward to it. Also check out this really cool experiment with a liquid: